The KSBA moduli space of stable log Calabi-Yau surfaces

Abstract

We prove that every irreducible component of the coarse Kollár-Shepherd-Barron and Alexeev (KSBA) moduli space of stable log Calabi--Yau surfaces admits a finite cover by a projective toric variety. This verifies a conjecture of Hacking-Keel-Yu. The proof combines tools from log smooth deformation theory, the minimal model program, punctured log Gromov-Witten theory and mirror symmetry.

93 pages, 10 figures. Final version accepted for publication in Forum of Mathematics, Pi